Loci with the complex logarithm and exponential

Problem

a) For which $z$ is $\ln(z)$ purely imaginary? Draw the curve solving $\operatorname{Im}\big(\ln z\big)=\dfrac{\pi}{6}$. b) For which $z$ is $e^{z}$ real positive, real negative, imaginary, or has real part equal to imaginary part? c) What does the circle $|z|=1$ become under the function $\ln(z)$?

Solution

Write $\ln z=\ln|z|+i\arg z$ and $e^{z}=e^{x}(\cos y+i\sin y)$ with $z=x+iy$. a) $\ln z$ is purely imaginary $\Leftrightarrow \operatorname{Re}(\ln z)=\ln|z|=0\Leftrightarrow |z|=1$: the unit circle. Moreover $\operatorname{Im}(\ln z)=\arg z=\dfrac{\pi}{6}$ is the ray from the origin with inclination $\dfrac{\pi}{6}$. b) $e^{z}$ real positive $\Leftrightarrow y=2k\pi$; real negative $\Leftrightarrow y=\pi+2k\pi$; imaginary $\Leftrightarrow y=\dfrac{\pi}{2}+k\pi$; real part $=$ imaginary part $\Leftrightarrow \cos y=\sin y\Leftrightarrow y=\dfrac{\pi}{4}+k\pi$. c) On $|z|=1$ we have $\ln z=\ln 1+i\arg z=i\arg z$ with $\arg z\in(-\pi,\pi]$: the circle becomes the segment of the imaginary axis from $-i\pi$ to $i\pi$. $\boxed{\,a)\ |z|=1\ \text{and ray}\ \arg z=\tfrac{\pi}{6}\,;\quad c)\ \text{segment}\ [-i\pi,\,i\pi]\,}$